J. Austral. Math. Soc. Sen B 36(1994), 107-116
CONTROLLABILITY OF NONLINEAR NEUTRAL VOLTERRA INTEGRODIFFERENTIAL SYSTEMS K. BALACHANDRAN and P. BALASUBRAMANIAM 1
(Received 2 July 1992; revised 24 February 1993) Abstract Sufficient conditions for the controllability of nonlinear neutral Volterra integrodifferential systems with implicit derivative are established. The results are a generalisation of the previous results, through the notions of condensing map and measure of noncompactness of a set.
1. Introduction Compartmental models are frequently used in theoretical epidemiology, physiology, and population dynamics to describe the evolution of systems which can be divided into separate compartments, marking the pathways of material flow between compartments and the possible outflow into the inflow from the environment of the system. Generally, the time required for the material flow between compartments cannot be neglected. A model for such system can be visualised as one in which compartments are connected by pipes which material passes through in definite time. Because of the time lags caused by pipes, the model equations for such systems are differential equations with deviating arguments, as opposed to the classical case where model equations are ordinary differential equations. For more details, we refer the reader to [1, 17,21]. A concrete example is the radiocardiogram, where the two compartments correspond to the left and right ventricles of the heart, and the pipes between them represent the pulmonary and systematic circulation. Pipes coming out from and returning into the same compartment may represent shunts and the coronary circulation [see 16]. A more simplified equation representing this model is a nonlinear neutral Volterra integrodifferential equation as in [18]. The aim of this paper is to study the controllability problem for such systems. 'Department of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamil Nadu, India © Australian Mathematical Society, 1994, Serial-fee code 0334-2700/94 107
108
K. Balachandran and P. Balasubramaniam
[2]
Several authors [11, 14, 19, 25] have studied the theory of functional differential equations. The problem of controllability of linear neutral systems has been investigated in [7, 20]. Motivation for physical control systems and its importance in other fields can be found in [19, 22]. Angell [2] and Chukwu [9] discussed the functional controllability of nonlinear neutral systems and Underwood and Chukwu [26] studied the null controllability for such systems. Further, Chukwu [10] considered the Euclidean controllability of a neutral system with nonlinear base. Onwuatu [23] discussed the problem for nonlinear systems of neutral functional differential equations with limited controls. Gahl [15] derived a set of sufficient conditions for the controllability of nonlinear neutral systems through the fixed point method developed by Dauer [13]. However Dacka [12] introduced a new fixed-point method of analysis to study the controllability of nonlinear systems with implicit derivative based on the measure of noncompactness of a set and Darbo's theorem. This method has been extended to a larger class of dynamical systems by Balachandran [4, 5]. Anichini etal. [3] studied the problem through the notions of condensing map and measure of noncompactness of a set. They used the fixed-point theorem due to Sadovskii. In this paper, we shall study the controllability of nonlinear neutral Volterra integrodifferential systems with implicit derivative, by suitably adopting the technique of Anichini et al. [3]. The results generalise those results obtained by Balachandran [6] where the nonlinear function / is independent of x.
2. Mathematical preliminaries We first summarise some facts concerning condensing maps; for definitions and results about the measure of noncompactness and related topics, the reader can refer to the paper of Dacka [12]. Let X be a subset of a Banach space. An operator T : X —• X is called condensing if, for any bounded subset E in X with /x(£) ^ 0, we have IA(T(E)) < /x(E), where lx(E) denotes the measure of noncompactness of the set E. We observe that, as a consequence of the properties of /x, if an operator T is the sum of a compact operator and condensing operator, then T itself is a condensing operator. Further, if the operator P : X - • X satisfies the condition \Px — Py\ < k\x — y\ for x, y € X, with 0 < k < 1, then the operator P is a fi - contractive operator with constant k: that is, /x(T(£)) < kfi(E) for any bounded set E in X. In this case, P has a fixed-point property (Sadovskii [24]). However, the condition | Px — Py \ < \x — y \ (x, y e X) is insufficient to ensure that P is a condensing map or that P will admit a fixed point (Browder [8]). The fixed-point property holds in the condensing case (Sadovskii [24]). Let Cn(I) denote the space of continuous /?"-valued functions on the interval /.
[3]
Controllability of nonlinear neutral volterra integrodifferential systems
109
For x 0, let 6(x,h)
= sup{\x(t) -x(s)\
:t,s
€ /
with \t - s\
R+ that arerightcontinuous and nondecreasing such that co(r) < r, for r > 0. Let / = [t0, tt]. LEMMA 1. [24] Let X c Cn(I) and let fl and y be functions defined on [0, tx — to] such that lim^o P(s) = lim^o Y(s) = 0- Ifa mapping T : X -> C n (/) is given such that it maps bounded sets into bounded sets, and is such that
9(T(x),h)
(e(x,P(h))) + y(h)
for all h e [0,h -to]andx
eX
with co G fi, then T is a condensing mapping. LEMMA 2. [3, 24] Let X c Cn(J), let J=[0,l], and let S C X be a bounded closed convex set. Let H : J x S —> X be a continuous operator such that, for any a e / , the map //(or, •) : S —> X is condensing -. If x ^ H(a, x) for any a e / and any x 6 35 (the boundary of S), then / / ( I , . ) has a fixed point. Finally it is possible to show that, for any bounded and equicontinuous set E in C\ (/), the following relation holds:
fiq(E) = ME) = n(DE) = Hc.{DE), where DE = {x : x € E). 3. Main result Consider the nonlinear neutral Volterra integrodifferential system
j t \xit) - j
C(t- s)x(s)ds - g(t)]
= Ax(t)+
f/ G(t-s)x(s)ds Git-
Jo
+ B(t)u(t) + f(t,x(t),x(t),u(t)),
(1)
where x e R", u e Rm, C(t) is an n x n continuously differentiable matrix valued function, G(t) is an n x n continuous matrix, and B(t) is a continuous n x m matrix valued function, A a constant n x n matrix and / and g are respectively continuous and continuously differentiable vector functions. Here the control functions are continuous.
110
K. Balachandran and P. Balasubramaniam
[4]
The solution of the system (1) can be written as [27]:
x{t) = Z(t) [x(0) - g(0)] + g(t) + f Z(t- s)g(s)ds Jo
+f
Z(t - s)[B(s)u(s) +
f(s,x(s),x(s),u(s)))ds,
Jo
where Z(t) is an n x n continuously differentiable matrix satisfying
^ I Z(r) - f C(t- s)Z(s)ds\ = AZ(t) + f G(t- s)Z(s)ds
(2)
with Z(0) = identity matrix. We say that system (1) is controllable if, for every x(0), X\ e R", there exists a control function u defined on / = [0, tt] such that the solution of (1) satisfies x(ti) = X\. Define the controllability matrix W(fl, h)=
[' Z(r, - s)B(s)B*(s)Z*(tl - s)ds, Jo
where the star denotes the matrix transpose. THEOREM. Suppose that the continuity condition on the matrices G, B, f and the continuous differentiability of C, g are satisfied for the system with the following additional conditions: (i)
\f(t,x,y,u)\
hm sup — r
IxKoo
(ii)
= 0;
he
there exists a continuous nondecreasing function co : R+ -> R+,withco(r) < r, such that \f(t,x,y,u)-f(t,x,z,u)\
=/" Jo
= f ' e~4s [4e2i"~s) - 4e 
